Subbalakshmi Lakshmipathy College of Science. Department of Mathematics

Transcription

1 ॐ Subbalakshmi Lakshmipathy College of Science Department of Mathematics As are the crests on the hoods of peacocks, As are the gems on the heads of cobras, So is Mathematics, at the top of all Sciences. Class II B.Sc., CS (Third Semester) Subject Mathematics Paper Numerical Methods With effect from (Under CBCS) SYLLABUS LESSON PLAN QUESTION PATTERN BLUE PRINT MODEL QUESTION PAPER QUESTION BANK ASSIGNMENT / TEST SCHEDULE Prepared By S. Mahaalingam 1

2 Semester III SYLLABUS NUMERICAL METHODS (Problem Oriented Paper) Unit 1 Numerical Solution of Algebraic and Transcendental Equations The Bisection Method - Iteration Method Condition for the convergence & Order of convergence of Iteration Method - Regula Falsi Method Newton-Raphson Method Criterion for the Convergence & Order of Convergence of Newton s Method - Horner s Method Unit Equations Solution of Simultaneous Linear Algebraic Direct Methods: Gauss-Elimination Method Gauss- Jordon Method Iterative Methods: Condition of Convergence of Iterative Methods Gauss Jacobi Method Gauss Seidel Method Unit 3 Interpolation with Equal Intervals Gregory-Newton Forward Interpolation Formula - Gregory- Newton Backward Interpolation Formula Gauss Forward Interpolation Formula Gauss Backward Interpolation Formula Stirling s Formula - Bessel s Formula Unit 4 Interpolation with Unequal Intervals: Divided Differences Newton s Interpolation Formula - Lagrange s Interpolation Formula Inverse Interpolation Numerical Integration: Trapezoidal Rule Simpson s 1/3 rd Rule - Simpson s 3/8 th Rule - Weddle s Rule

3 Unit 5 Solutions of Ordinary Differential Equations of the First Order of the Form y = f(x, y) with y(x o ) = y o Solution by Taylor Series (Type I) Euler s Method Improved Euler Method Modified Euler Method Runge- Kutta Method of Fourth Order Only Text Book Numerical Methods by P. Kandasamy, K. Thilagavathy, K. Gunavathy Published by S. Chand & Company Limited, Ram Nagar, New Delhi Chapters Unit 1 Chapter 3 ( ) Unit Chapter 4 (4.1-4., ) Unit 3 Chapter 6 ( ), 7 ( ) Unit 4 Chapter 8 (8.1-8., ), 9 ( , ) Unit 5 Chapter 11(11.5, ) References Numerical Methods in Engineering and Science by Dr. B.S. Grewal Published by Khanna Publishers, -B Nath Market, Nai Sarak, Delhi Numerical Methods by S. Kalavathy, (Publisher-Thomson), Copublished by Vijay Nicole Imprints Private Ltd., C-7, Nelson Chambers, 115, Nelson Manickam Road, Chennai 9. 3

4 SUBBALAKSHMI LAKSHMIPATHY COLLEGE OF SCIENCE Department of Mathematics LESSON PLAN Class : II Year B.Sc., Computer Science Semester : Third Subject : Numerical Methods (Code CBCS 301) No. of Hours : 80 Uni Chapt ers Topics 01 Numerical Solution of Algebraic & Numb er of Hours Transcendental Equation 01 Preliminary Concepts 1 0 The Bisection Method 03 Iteration Method 3 4

5 04 Condition of Convergence & Order of Convergence 05 Regula Falsi Method 06 Newton-Raphson Method 3 07 Criterion for the Convergence & Order of 1 Convergence 08 Horner s Method 3 0 Solution of Simultaneous Linear Algebraic Equations 01 Gauss Elimination Method 0 Gauss Jordon Method 3 03 Condition of Convergence of Iteration Methods 1 04 Gauss Jacobi Method 3 05 Gauss Seidel Method 3 06 Simple Problems 4 03 Interpolation with Equal Intervals 01 Introduction and Difference Table 0 Gregory-Newton Forward Interpolation Formula 03 Gregory-Newton Backward Interpolation Formula 04 Gauss Forward Interpolation Formula 05 Gauss Backward Interpolation Formula 06 Stirling s Formula for Central Interpolation 3 07 Bessels s Formula for Central Interpolation 3 04 Interpolation with Unequal Intervals 01 Divided Difference 1 0 Newton s Divided Difference Method 3 03 Lagrange s Interpolation Formula 3 04 Lagrange s Formula for Inverse Interpolation 1 Numerical Integration 05 Introduction 1 06 Trapezoidal Rule for Numerical Integration 1 07 Simpson s 1/3 rd Rule 08 Simpson s 3/8 th Rule 09 Weddle s Rule 1 05 Solutions of Ordinary Differential Equations of the First Order 01 Solution of First Order DE by Taylor Series (Type I) 3 5

6 0 Solution of First Order DE by Euler s Method 03 Solution of First Order DE by Improved Euler s 3 Method 04 Solution of First Order DE by Modified Euler s 3 Method 05 Solution of First Order DE by Runge-Kutta 3 Method of 4 th Order 06 Simple Problems SUBBALAKSHMI LAKSHMIPATHY COLLEGE OF SCIENCE Department of Mathematics EXTERNAL QUESTION PATTERN Class : II B.Sc., Computer Science Max. Marks : 75 Subject : Numerical Methods Duration : 3 hours Subject Code : CBCS 301 Part A Answer all the questions (10 x 1) 10 Part B Answer all the questions (Either or type) (5 x 7) 35 Part C Answer any THREE questions (3 out of 5) (3 x 10) 30 6

7 Total Marks 75 INTERNAL QUESTION PATTERN Class : II B.Sc., Computer Science Max. Marks : 30 Subject : Numerical Methods Duration : hours Subject Code : CBCS 301 Part A Answer all the questions (6 x 1) 06 Part B Answer all the questions (Either or type) ( x 7) 14 Part C Answer any THREE questions (3 out of 5) (1 x 10) 10 Total Marks 30 BLUE PRINT Part A Part B Part C 1. Unit 16. Unit Unit 1 / Unit Unit 1. Unit 17. Unit 4 1. Unit / Unit 17. Unit 3. Unit 8. Unit Unit 3 / Unit Unit 3 4. Unit 9. Unit Unit 4 / Unit Unit 4 7

8 5. Unit Unit Unit 5 / Unit 5 0. Unit 5 Directions for the Examiners The students having this paper didn t study Mathematics in depth at higher secondary levels. Moreover, they are doing B.Sc., Computer Science and not B.Sc., Mathematics. So, just problematic approach is given for students. Direction for the Students Scientific / Engineering calculators and Clerk s Table are allowed in both the internal and the external examinations. SUBBALAKSHMI LAKSHMIPATHY COLLEGE OF SCIENCE Department of Mathematics MODEL QUESTION PAPER Class : II B.Sc., Computer Science Semester : Third 8

9 Subject : Numerical Methods Max. Marks : 75 Subject Code : CBCS 301 Duration : 3 hours Part A (Answer all the questions) 10 x 1 = If + 3 is a root of an equation, then find the other roots.. What is a reciprocal equation? 3. What is the order of convergence of Newton s method? 4. Write the condition of convergence of Iteration method. 5. What is the condition for existence of inverse, of a square matrix A? 6. Which method uses the process of back substitution in solving simultaneous linear algebraic equations? 7. What is the relation between the forward and the shift operator? 8. When we apply Stirling s central difference interpolation formula? 9. Write trapezoidal rule for numerical integration. 10. Write Euler s formula to solve a first order differential equation. Part B 5 x 7 = 35 (Answer all the questions) 11. (a) Solve the reciprocal equation x 5-5x 4 + 9x 3-9x + 5x - 1 = 0. (OR) (b) If α, β, γ are the roots of x 3 + qx + r = 0, find the value of (α + β)(β + γ)(γ + α) and (α + β - γ)(β + γ - α)(γ + α - β) 9

10 1. (a) Find a real root of the equation cos x = 3x 1 correct to 4 places of decimals by iteration method. (OR) (b) Find the positive root of x 3 x = 1 correct to four decimal places by bisection method. 13. (a) Solve the system of equations x + y + z = 3, x + 3y + 3z = 10, 3x y + z = 13 by Gauss Elimination method. (OR) (b) Solve the following equations using relaxation method: 10x - y - z = 6, -x + 10y - z = 7, -x y + 10z = (a) Use Newton Gregory forward difference formula to calculate 5.5 given 5 =.36, 6=.449, 7=.646 and 8=.88. (OR) (b) Using Newton s divided difference formula, find f () given the following table: x : f (x) : (a) Evaluate x (OR) 6 using Simpson s rule. (b) Using Taylor series method, find, correct to four decimal dy places, the value of y (0.1), given = x +y and y (0) = 1. Part C (Answer any TWO questions) x 15 = Find the condition that the roots of the equation x 3 + px + qx + r = 0 may be in (a) Arithmetic Progression (b) Geometric Progression (c) Harmonic Progression 10

11 17. Find the positive root of x 3 + 3x 1 = 0, correct to decimal places, by Horner s method. 18. Solve the system of equations 10x - 5y - z = 3, 4x - 10y + 3z = -3, x + 6y + 10z = -3 by Gauss Seidel method. 19. Use Gauss s forward formula to get y 30 given that, y 1 = , y 5 = , y 9 = , y 33 = , y 37 = Apply the fourth order Runge-Kutta method to find y (0.) given y = x + y, y (0) = 1 by taking h = 0.1. QUESTION BANK SUBJECT NUMERICAL METHODS CODE CBCS 301 CLASS II B.Sc., Computer Science (Third Semester) UNIT 1 Equations PART A The Solution of Numerical Algebraic and Transcendental 1. Define an algebraic equation of n th degree.. Diminish by 3 the roots of x 4 + 3x 3 - x - 4x - 3 = Increase by the roots of x 4 - x 3-10x + 4x + 4 = Diminish the roots of 3x 3 + 8x + 8x + 1 = 0 by Increase by 7 the roots of the equation 3x 4-7x 3-15x + x - = State Descarte s rule of sings. 7. Determine the nature of the roots of x 6 + 3x 5 + 5x - 1 = 0. 11

12 8. Discuss the nature of roots of the equation x 5 + x 4 + x + x + 1 = Discuss the nature of roots of the equation x 6-3x 4 + x 3-1 = Discuss the nature of roots of the equation x 5 + 5x 9 = Discuss the nature of roots of the equation x 5 + x 1 = Discuss the nature of roots of the equation x = Discuss the nature of roots of the equation x 5 6x 4x + 7 = Write the condition of convergence of Newton s method. 15. Write the condition of convergence of Iteration method. 16. What is the order of convergence of Newton s method? 17. What is the order of convergence of Iteration method? 18. Write the iterative formula of Newton s method. 19. Write the iterative formula of Iteration method. 0. Write the formula to compute c in Bisection method. 1. Write the formula to compute c in Regula-falsi method.. What is the other name of Newton s method? 3. Obtain the interval of the positive root of x 3 x 1 = Write the Iterative formula and condition of convergence of Newton s method. 5. Write the Iterative formula and condition of convergence of Iteration method. 6. Write short notes on Horner s method. 7. Find an iterative formula to find N 8. Find an iterative formula to find the reciprocal of a given number N. 9. Find the order of convergence of Newton s method. 30. Find the order of convergence of Iterative method. 31. Construct φ(x) from cos x = 3x 1 that satisfies the condition of convergence. 3. Construct φ(x) from 4x - e x = 0 that satisfies the condition of convergence. 1

13 33. How will you find the negative real root of the equation f(x) = 0? PART B 34. Explain bisection method to find the real root of the equation f(x) = Find the positive root of x 3 x = 1 correct to four decimal places by bisection method. 36. Assuming that a root of x 3 9x + 1 = 0 lies in the interval (, 4), find that root by bisection method. 37. Find the positive root of x cos x = 0 by bisection method. 38. Find the positive root of x 4 x 3 x 6x 4 = 0 by bisection method correct to places of decimals. 39. Find the positive root of x 3 = x + 5 by false position method. 40. Solve for a positive root of x 3 4x + 1 = 0 by regula falsi method. 41. Find an approximate root of x log 10 x 1. = 0 by false position method. 4. Solve for a positive root of x cos x = 0 by regula falsi method. 43. Solve the equation x tan x = -1 by regula falsi method, starting with a =.5 and b = 3 correct to 3 decimal places. 44. Find a positive root of x e x = by the method of false position. 45. Find the positive root of the equation x 3 4x 9 = 0 by bisection method. 46. Find the positive root of the equation e x = 3x by bisection method. 47. Find the positive root of the equation x 3 + 4x 1 = 0 by bisection method. 13

14 48. Find the positive root of the equation 3x = cos x + 1 by bisection method. 49. Find the positive root of the equation x 3 + x 1 = 0 by bisection method. 50. Find the positive root of the equation x = 3 + cos x by bisection method. 51. Solve x e x = 3 for a positive root by false position method. 5. Solve 4x = e x for a positive root by false position method. 53. Solve x log 10 x = 1. for a positive root by false position method. 54. Solve e -x = sin x for a positive root by false position method. 55. Solve x 3 5x 7 = 0 positive root by false position method. 56. Solve x log 10 x = 7 for a positive root by false position method. 57. Solve 3x cos x = 1 for a positive root by false position method. 58. Solve x 3log x = 5 for a positive root by false position method. 59. Find the positive root of f(x) = x 3 3x 6 = 0 by Newton Raphson method correct to five decimal places. 60. Using Newton s method, find the root between 0 and 1 of x 3 = 6x 4 correct to five decimal places. 61. Find the real positive root of 3x cos x 1 = 0 by Newton s method correct to 6 decimal places. 6. Find the positive root of x = cos x by Newton s method. 63. Find the root of 4x e x = 0 that lies between and 3 using Newton s method. 64. Find the positive root of x 3 x 1 = 0 using Newton-Raphson method. 65. Find the positive root of f(x) = cos x x e x using Newton-Raphson method. 66. Find a real root of x 3 + x + 50x + 7 = 0 using Newton-Raphson method. 14

15 67. Find the value of 31 1 using Newton-Raphson method. 68. Find the positive root of x sin x = 0 by Newton s method. 69. Find a real root of x 3 + x + 1 = 0 using Newton-Raphson method. 70. Find the cube root of 4 correct to 3 decimal places using Newton s method. 71. Find a real root of x 3 x = 0 using Newton-Raphson method. 7. Solve for a positive root of x e x = 1 using Newton-Raphson method. 73. Solve for a positive root of x 3 sin x = 5 using Newton- Raphson method. 74. Find a positive root of x e x = cos x using Newton-Raphson method. 75. Solve e x 3x = 0 by the method of iteration. 76. Find a real root of the equation cos x = 3x 1 correct to 4 places of decimals by iteration method. 77. Solve the equation x 3 + x 1 = 0 for the positive root by iteration method. 78. Solve x 3 = x + 5 for the positive root by iteration method. 79. Find a positive root of 3x - 1+ sin x = 0 by iteration method. 80. Solve the equation 3x cos x = 0 using iteration method. 81. Solve the equation x 3 + x + 1 = 0 using iteration method. 8. Solve the equation x log 10 x = 7 using iteration method. 83. Solve the equation cos x = 3x - 1 using iteration method. PART C 84. Using bisection method, find the negative root of x 3 4x + 9 = Find a positive root of 3x = 1+ sin x by bisection method. 15

16 86. Solve x 3 + x + 10x -0 for a positive root by false position method. 87. Find a negative real root of sin x = 1 + x 3 using Newton Raphson method. 88. Find a negative real root of x + 4 sin x = 0 using Newton Raphson method. 89. Solve for x from cos x x e x = 0 by iteration method. 90. Find an iterative formula to find N (where N is a positive number) and hence find Find an iterative formula to find the reciprocal of a given number 1 N and hence find the value of Find the positive root of x 3 + 3x 1 = 0, correct to two decimal places, by Horner s method. 93. Find the positive root between 1 and which satisfies x 3 3x + 1 = 0 to 3 decimal places by using Horner s method. 94. By Horner s method, find the root of x 3 3x +.5 = 0 that lies between and By Horner s method, find the root of x 3 + 6x + =0 that lies between 0 & By Horner s method, find the root of x 3 x 3x - 4 = 0 whose root lies between 3 and By Horner s method, find the positive root of x 3 8x - 40 = By Horner s method, find the positive root of h 3 + 3h 1h - 11 = By Horner s method, find the positive root of x 3 6x - 18 = By Horner s method, find the positive root of x 3 x = By Horner s method, find the root of 10x 3 15x + 3 = 0 that lies between 1 and. 16

17 THINGS TO REMEMBER 17

18 UNIT PART A Solution of Simultaneous Linear Algebraic Equations 1. Name the two types of methods involved in solving simultaneous linear algebraic equations.. Name any two direct methods to solve simultaneous linear algebraic equations. 3. What is the condition for existence of inverse, of a square matrix A? 4. Name any two iterative methods to solve simultaneous linear algebraic equations. 5. Which method uses the process of back substitution in solving simultaneous linear algebraic equations? 6. Give the conditions of converge of Iterative methods to solve system of simultaneous linear equations. 7. What is the major difference between direct and iterative methods in solving system of simultaneous linear equations? 8. Check whether the system of equations x - 3y + 10z = 3, -x + 4y + z = 0, 5x + y + z = -1 satisfies the condition of convergence or not. 9. Check whether the system of equations 8x + y + z + w = 14, x + 10y + 3z + w = -8, x - y - 0z + 3w = 111, 3x + y + z + 19w = 53 satisfies the condition of convergence or not. 10. When a matrix is said to be diagonally dominant? 18

19 11. Write Gauss Jacobi iterative formula to solve the system of equations a 1 x + b 1 y + c 1 z = d 1, a x + b y + c z = d, a 3 x + b 3 y + c 3 z = d Write Gauss Jacobi iterative formula to solve the system of equations a 1 x + b 1 y + c 1 z = d 1, a x + b y + c z = d, a 3 x + b 3 y + c 3 z = d Write the condition of convergence of Relaxation method. 14. Write the condition of convergence of Gauss Jacobi method. 15. Write the condition of convergence of Gauss Seidel method. PART B 16. Solve the system of equations x + y + z = 3, x + 3y + 3z = 10, 3x y + z = 13 by Gauss Elimination method. 17. Solve the system of equations x + 3y - z = 5, 4x + 4y - 3z = 3, x 3y + z = by Gauss Elimination method. 18. Using Gauss Elimination method, solve the system 3.15x 1.96y z = 1.95,.13x + 5.1y.89z = -8.61, 5.93x y +.15z = Solve by Gauss Elimination method: 3x + 4y + 5z = 18, x - y + 8z = 13, 5x + 7z = Solve the system of equations x - y + z = 1, -3x + y - 3z = -6, x 5y + 4z = 5 by Gauss Elimination method. 1. Solve the system of equations x + 3y + 10z = 4, x + 17y + 4z = 35, 8x + 4y - z = 3 by Gauss Elimination method. 19

20 . Solve the system of equations x - 3y - z = -30, x - y - 3z = 5, 5x y - z = 14 by Gauss Elimination method. 3. Solve the system of equations 10x + y + z = 1, x + 10y + z = 1, x + y + 10z = 1 by Gauss Elimination method. 4. Solve the system of equations 3x + y - z = 3, x - 8y + z = -5, x y + 9z = 8 by Gauss Elimination method. 5. Solve the system of equations 3x - y + z = 1, x + y + 3z = 11, x y - z = by Gauss Elimination method. 6. Solve the system of equations x - 3y + z = -1, x + 4y + 5z = 5, 3x 4y + z = by Gauss Elimination method. 7. Solve the system of equations x + y + 3z = 6, x + 4y + z = 7, 3x + y + 9z = 14 by Gauss Elimination method. 8. Solve the system of equations 4x + y + 3z = 11, 3x + 4y + z = 11, x + 3y + z = 7 by Gauss Elimination method. 9. Solve the system of equations x + y + z = 4, 3x + y - 3z = -4, x - 3y - 5z = -5 by Gauss Elimination method. 30. Solve the system of equations x + 6y - z = -1, 5x - y + z = 11, 4x y + 3z = 10 by Gauss Elimination method. 31. Solve the system of equations 6x - y + z = 13, x + y + z = 9, 10x + y - z = 19 by Gauss Elimination method. 3. Solve the system of equations x + 4y + z = 3, 3x + y - z = -, x y + z = 6 by Gauss Elimination method. PART C 33. Solve the system of equations x + y + z = 3, x + 3y + 3z = 10, 3x y + z = 13 by Gauss Jordon method. 34. Apply Gauss Jordon method to find the solution of the following system 10x + y + z = 1, x + 10y + z = 13, x + y + 5z = 7. 0

21 35. Solve the system of equations x - y + z = 1, -3x + y - 3z = -6, x 5y + 4z = 5 by Gauss Jordon method. 36. Solve the system of equations x + 3y + 10z = 4, x + 17y + 4z = 35, 8x + 4y - z = 3 by Gauss Jordon method. 37. Solve the system of equations x - 3y - z = -30, x - y - 3z = 5, 5x y - z = 14 by Gauss Jordon method. 38. Solve the system of equations 10x + y + z = 1, x + 10y + z = 1, x + y + 10z = 1 by Gauss Jordon method. 39. Solve the system of equations 3x + y - z = 3, x - 8y + z = -5, x y + 9z = 8 by Gauss Jordon method. 40. Solve the system of equations 3x - y + z = 1, x + y + 3z = 11, x y - z = by Gauss Jordon method. 41. Solve the system of equations x - 3y + z = -1, x + 4y + 5z = 5, 3x 4y + z = by Gauss Jordon method. 4. Solve the system of equations x + y + 3z = 6, x + 4y + z = 7, 3x + y + 9z = 14 by Gauss Jordon method. 43. Solve the system of equations 4x + y + 3z = 11, 3x + 4y + z = 11, x + 3y + z = 7 by Gauss Jordon method. 44. Solve the system of equations x + y + z = 4, 3x + y - 3z = -4, x - 3y - 5z = -5 by Gauss Jordon method. 45. Solve the system of equations x + 6y - z = -1, 5x - y + z = 11, 4x y + 3z = 10 by Gauss Jordon method. 46. Solve the system of equations 6x - y + z = 13, x + y + z = 9, 10x + y - z = 19 by Gauss Jordon method. 47. Solve the system of equations x + 4y + z = 3, 3x + y - z = -, x y + z = 6 by Gauss Jordon method. 48. Solve the system of equations 5x 1 + x + x 3 + x 4 = 4, x 1 + 7x + x 3 + x 4 = 1, x 1 + x + 6x 3 + x 4 = -5, x 1 + x + x 3 + 4x 4 = -6 by Gauss Jordon method. 1

22 49. Solve the system of equations x + y + z w = -, x + 3y - z + w = 7, x + y + 3z w = -6, x + y + z + w = by Gauss Elimination method. 50. Solve the system of equations w + x + y + z =, w - x + y - z = -5, 3w + x + 3y + 4z = 7, w - x - 3y + z = 5 by Gauss Jordon method 51. Solve the system of equations x + y + z w = -, x + 3y - z + w = 7, x + y + 3z w = -6, x + y + z + w = by Gauss Jordon method. 5. Solve the system of equations 10x - 5y - z = 3, 4x - 10y + 3z = -3, x + 6y + 10z = -3 by Gauss Jacobi method. 53. Solve the system of equations 8x - 3y + z = 0, 4x - 10y + 3z = -3, x + 6y + 10z = -3 by Gauss Jacobi method. 54. Solve the system of equations x + y + 54z = 110, 7x + 6y - z = 85, 6x + 15y + z = 7 by Gauss Jacobi method. 55. Solve the system of equations 5x - y + z = -4, x + 6y - z = -1, 3x + y + 5z = 13 by Gauss Jacobi method. 56. Solve the system of equations 8x + y + z = 8, x + 4y + z = 4, x + 3y + 3z = 5 by Gauss Jacobi method. 57. Solve the system of equations 30x - y + 3z = 75, x + y + 18z = 30, x + 17y - z = 48 by Gauss Jacobi method. 58. Solve the system of equations 10x - y + z = 1, x + 9y - z = 10, x - y + 11z = 0 by Gauss Jacobi method. 59. Solve the system of equations 8x - y + z = 18, x + 5y - z = 3, x + y - 3z = -16 by Gauss Jacobi method. 60. Solve the system of equations 4x + y + z = 14, x + 5y - z = 10, x + y + 8z = 0 by Gauss Jacobi method. 61. Solve the system of equations x - y + 10z = 30.6, x + 5y - z = 10.5, 3x + y + z = 9.3 by Gauss Jacobi method. 6. Solve the system of equations 83x + 11y - 4z = 95, 7x + 5y + 13z = 104, 3x + 8y + 9z = 71 by Gauss Jacobi method.

23 63. Solve the system of equations 10x - 5y - z = 3, 4x - 10y + 3z = -3, x + 6y + 10z = -3 by Gauss Seidel method. 64. Solve the system of equations 8x - 3y + z = 0, 4x - 10y + 3z = -3, x + 6y + 10z = -3 by Gauss Seidel method. 65. Solve the system of equations 8x + 4y - z = 3, x + 3y + 10z = 4, x + 17y + 4z = 35 by Gauss Seidel method. 66. Solve the system of equations x + y + 54z = 110, 7x + 6y - z = 85, 6x + 15y + z = 7 by Gauss Seidel method. 67. Solve the system of equations 5x - y + z = -4, x + 6y - z = -1, 3x + y + 5z = 13 by Gauss Seidel method. 68. Solve the system of equations 8x + y + z = 8, x + 4y + z = 4, x + 3y + 3z = 5 by Gauss Seidel method. 69. Solve the system of equations 30x - y + 3z = 75, x + y + 18z = 30, x + 17y - z = 48 by Gauss Seidel method. 70. Solve the system of equations 10x - y + z = 1, x + 9y - z = 10, x - y + 11z = 0 by Gauss Seidel method. 71. Solve the system of equations 8x - y + z = 18, x + 5y - z = 3, x + y - 3z = -16 by Gauss Seidel method. 7. Solve the system of equations 4x + y + z = 14, x + 5y - z = 10, x + y + 8z = 0 by Gauss Seidel method. 73. Solve the system of equations x - y + 10z = 30.6, x + 5y - z = 10.5, 3x + y + z = 9.3 by Gauss Seidel method. 74. Solve the system of equations 83x + 11y - 4z = 95, 7x + 5y + 13z = 104, 3x + 8y + 9z = 71 by Gauss Seidel method. THINGS TO REMEMBER _ 3

24 4

25 UNIT 3 Interpolation with equal intervals PART A 1. What is interpolation and extrapolation?. Write the formula for forward and backward difference operator. 3. Write the formula for the shift operator. 4. Write the formula for central difference operator. 5. What is the relation between the forward and the shift operator? 6. Give any two central difference interpolation methods. 7. What is the relation between Gauss and Stirling s formula? 8. When Gauss forward formula is useful for interpolation? 9. When Gauss backward formula is useful for interpolation? 10. When we apply Stirling s central difference interpolation formula? 11. When Stirling s formula is useful for interpolation? 1. When Bessel s formula is useful for interpolation? 13. What is the other name of Bessel s formula? 14. Write Newton Gregory forward difference interpolation formula. 5

26 15. Write Newton Gregory backward difference interpolation formula. 16. Write Gauss forward difference interpolation formula. 17. Write Gauss backward difference interpolation formula. 18. Write Stirling s central difference interpolation formula. 19. Write Bessel s central difference interpolation formula. 0. What is the error in polynomial interpolation? 1. What is the error in Newton s forward interpolation formula?. What is the error in Newton s backward interpolation formula? PART B 3. Find a polynomial of degree four which takes the values x : y : Find a polynomial of degree two which takes the values x : y : Find the missing value of the table given below. Year : Export (in tons) : Find the missing value of the following table. x : y : Find the annual premium at the age of 30, given Age : Premium : Find the polynomial of least degree passing through the points (0, -1), (1, 1), (, 1), (3, -). 6

27 9. Construct a polynomial for the data given below. Find also y (x = 5). x : y : Given the following data, express y as a function of x. x : y : From the table given below find f (3.4). x : y : Find f (.5) given, x : y : Find f (0.5), if f (-1) = 0, f (0) = 175, f (1) = 8, f () = Using Newton s backward formula, find the polynomial of degree 3 passing through (3,6), (4, 4), (5, 60), (6, 10) 35. Find the value of f (x) at x = 9 from the following table: x : f (x) : Calculate 5. 5 given 5 =.36, 6=.449, 7=.646 and 8= Derive Stirling s central difference interpolation formula for equal intervals. 38. Apply Gauss forward formula to obtain f (x) at x = 3.5 from the table below. x : f (x) : Given f () = 10, f (1) = 8, f (0) = 5, f (-1) = 10, find f (½) by Gauss forward formula. 40. Apply central difference formula to find f (1) given, x :

28 f (x) : From the table below, find y (5) given, using Bessel s formula. x : f (x) : Apply Bessel s formula to get the value of y 5 given y 0 = 854, y 4 = 316, y 8 = 3844, y 3 = Using Bessel s formula, estimate 3 ( 46.4) given, x : x : Apply Bessel s formula to get the value of y (45) given, x : y : Using Bessel s formula obtain the value of y (5) given, x : y : PART C 46. Find the values of y at x = 1 and x = 8 from the following data. x : y : From the following table of half-yearly premium for policies maturing at different ages, estimate the premium for policies maturing at age 46 and 63: Age x: Premium y: From the following table, find the value of tan 45 o 15. 8

29 x o : tan x o : The population of a town is as follows. Year x : Population in lakhs : Estimate the population increase during the period 1946 to From the following data, find θ at x = 43 and x = 84. x : θ : From the data given below, find the number of students whose weight is between 60 & 70. Weight in lbs : No. of students: The following data are taken from the steam table. Temp. o C : Pressure kgf/cm : Find the pressure at temperature t = 14 o and t = 175 o. 53. From the table given below, find sin 5 o by using Newton s forward interpolation formula. Also estimate the error. x : y=sin x : Find the value of e 1.85 given e 1.7 = , e 1.8 = , e 1.9 = , e.0 = , e.1 = 8.166, e. = 9.050, e.3 = Find log 10 π given, log = , log 3.14 = , log = , log = , log = where π =

30 56. Find the value of y at x = 1.05 from the table given below: x : y : Find the value of f (1.0) at x = 1.05 from the table given below: x : y : Find y at x = 105 from the following data: x : y : Find y (4) from the data given below. x : y : Find y (0.47) from the data given below. x : y : Find f (0.) if f (0) = 176, f (1) = 185, f () = 194, f (3) = 03, f (4) = 1, f (5) = 0, and f (6) = For the following data, find the forward and backward difference polynomials. Interpolate at x = 0.5 and x = x : f (x) : From the following table, find the value of tan 45 o 15. x o : tan x o : Estimate e -1.9 from the given data. x :

31 e x : The population of a town is given below. Estimate the population in the year 1895 & 195. Year x : Population (in 1000 s) : Find y (3) if y (10) = 35.3, y (15) = 3.4, y (0) = 9., y (5) = 6.1, y (30) = 3., y (30) = Estimate sin 38 o from the data given below. x : sin x : Find y (8) given, x : y : Find y (1.0) given, x : y : Apply Gauss s forward central difference formula and estimate f (3) from the table: x : y = f(x) : Using the following table, apply Gauss s forward formula to get f (3.75). x : f (x) : Using Gauss s backward interpolation formula find the population for the year 1936 given that Year :

32 Population (in thousand) : Find the value of cos 51 o 4 by using Gauss s backward interpolation formula from the table given below. x : 50 o 51 o 5 o 53 o 54 o y = cos x : If 1500 = , 1510 = , 150 = , 1530 = , find 1516 by Gauss s backward formula. 75. Use Gauss s interpolation formula to get y 16 given x : y : Use Gauss s forward formula to get y 30 given that, y 1 = , y 5 = , y 9 = , y 33 = , y 37 = Apply Gauss s backward formula to obtain sin 45 o given the table below: x o : sin x o : The values of e -x for various values of x are given below. Find e by Gauss forward formula. x : e -x : Using Stirling s formula, find y (1.) from the following table. x :

33 y : From the following table estimate e0.644 correct to five decimals using Stirling s formula. x : e x : From the following table estimate e0.644 correct to five decimals using Bessel s formula. x : e x : The following table gives the values of the probability integral f (x) = x π x e 0, for certain values of x. Find the value of this integral when x = using Stirling s formula. x : y= f (x) : The following table gives the values of the probability integral f (x) = x π x e 0, for certain values of x. Find the value of this integral when x = using Bessel s formula. 33

34 x : y= f (x) : Given the following table, find y (35) by using Stirling s formula and Bessel s formula. x : y : From the following table, using Stirling s formula, estimate the value of tan 16 o. x : 0 o 5 o 10 o 15 o 0 o 5 o 30 o y = tan x : Using Stirling s formula, estimate f (1.) from the following table. x : y = f (x) : Estimate 1. 1 using Stirling s formula from the following table. x : f (x) : Find f (1.) using Bessel s formula. x : f (x) : Using Bessel s formula find y (6.5) given, x : y (x) : Obtain the value of y (7.4) using Bessel s formula, given, 34

35 x : y (x) : From the table below, find f (.73) using Bessel s formula. x : f (x) : THINGS TO REMEMBER 35

36 _ UNIT 4 Interpolation with un-equal intervals and Numerical Integration PART A 1. Define the first divided difference.. State any two properties of a divided difference. 3. Write Newton s divided difference formula for unequal intervals. 4. Write Lagrange s formula for unequal intervals. 5. Write Trapezoidal rule for integration. 6. What is the order of error in Trapezoidal rule? 36

37 7. What is the truncation error in Trapezoidal rule? 8. Write Simpson s 1/3 rd rule for integration. 9. What is the order of error in Simpson s 1/3 rd rule? 10. What is the truncation error in Simpson s 1/3 rd rule? 11. When Simpson s 1/3 rd rule is applicable? 1. Write Simpson s 3/8 th rule for integration. 13. When Simpson s 3/8 th rule is applicable? 14. Write Weddle s rule for numerical integration. 15. When Weddle s rule is applicable? PART B 16. Find the equation y = f (x) of least degree and passing trough the points (-1, -1), (1, 15), (, 1), (3, 3). Find also y at x = 0 using Newton s divided difference formula. 17. From the following table find f (x) and hence f (6) using Newton s interpolation formula. x : f (x) : The following table gives same relation between steam pressure and temperature. Find the pressure at temperature 37.1 o using Newton s divided difference formula. T : 361 o 367 o 378 o 387 o 399 o P : Using the following table, find f (x) as a polynomial by using Newton s formula. x : f (x) : Find y (x = 0) using Newton s divided difference formula, given, 37

38 x : y (x) : Find a cubic polynomial of x given, x : f (x) : Find the third divided difference of f (x) with arguments, 4, 9, 10 where f (x) = x 3 x. 3. Given the data, find f (x) as a polynomial of degree by using divided difference method. x : 1-4 f (x) : Find a polynomial f (x) of lowest degree using divided difference method, which takes the values 3, 7, 9 and 19 when x =, 4, 5, Find log using Newton s divided difference method, given, x : log 10 x : From the following table, find f (5) using divided difference method. x : f (x) : Using divided difference table, find f (x) which takes the values 1, 4, 40, 85 as x = 0, 1, 3 and Using Lagrange s formula, prove y 1 = y (y 5 y -3 ) + 0. (y -3 y -5 ) nearly. 9. Using Lagrange s interpolation formula, find y (10) from the following table: x : y :

39 30. Using Lagrange s formula to fit a polynomial to the data. x : y : Use Lagrange s formula to find the parabola of the form y = ax + bx + c passing through the points (0, 0), (1, 1) and (, 0). 3. Find the value of θ given f (θ) = where f (θ) = θ dθ 1 1 sin 0 θ using the table. θ : 1 o 3 o 5 o f (θ) : From the table given below, find y (x = ), using Lagrange s formula. x : y : Use Lagrange s formula to find f (6) given, x : f (x) : If y 1 = 4, y 3 = 10, y 4 = 340, y 6 = 544, find y 5 using Lagrange s formula. 36. Find y (6) using Lagrange s formula, given, y (1) = 4, y () = 5, y (7) = 5, y (8) = Find y (10) using Lagrange s formula, given, y (5) = 1, y (6) = 13, y (9) = 14, y (11) = If y o = 1, y 3 = 19, y 4 = 49, y 6 = 181, then find y 5 using Lagrange s formula. 39. Find f (0) using Lagrange s method, given, x : -1-4 f (x) :

40 40. The following are the measurements t made on a curve recorded by the oscillograph representing a change of current i due to a change in the conditions of an electric current. t : i : Using Lagrange s formula, find I at t = Using Lagrange s formula, find x given y = 0.3 from the data. x : y : If log 300 =.4771, log 304 =.489, log 305 =.4843, log 307 =.4871, find log 301 using Lagrange s formula. 43. Use Lagrange s formula to find the value of x when y (x) = 19 given, x : 0 1 y : Given u o = -1, u 1 = 0, u 3 = 6, u 4 = 1, find u using Lagrange s formula. 45. If f (0) = 6, f (1) = 9, f (3) = 33, f (7) = -15, then find f () using Lagrange s formula. 46. Given f (30) = -30, f (34) = -13, f (38) = 3 and f (4) = 18. Find x so that f (x) = Use Lagrange s formula to find f (x), given the table: x : f (x) : Find the value of tan 33 o by using Lagrange s formula of interpolation given, x : 30 o 3 o 35 o 38 o tan x :

41 49. Use Lagrange s formula to find x corresponding to y = 85, given, x : f (x) : Using Lagrange s formula, fit a polynomial to the following data. x : y : Find the parabola passing through the points (0, 1), (1, 3) and (3, 55) using Lagrange s interpolation formula. 5. Use Lagrange s formula to find y () from the following table. x : y : Evaluate 1 + x 1 0 using Trapezoidal rule with h = 0.. Hence obtain the value of π. 54. From the following table, find the area bounded by the curve and the x-axis from x = 7.47 to x = 7.5. x : y = f (x) : x 55. Evaluate e by Simpson s one-third rule correct to five 0 decimal places, by proper choice of h. 56. A curve passes through the points (1, ), (1.5,.4), (.0,.7), (.5,.8), (3, 3), (3.5,.6) and (4.0,.1). Obtain the area bounded by the curve, the x-axis and x = 1 and x = 4. Also find the volume of solid of revolution got by revolving this area about the x-axis. 41

42 57. A river is 80 metres wide. The depth d in metres at a distance x metres from one bank is given by the following table. Calculate the area of cross-section of the river using Simpson s rule. x : d : The table below gives the velocity v of a moving particle at time t seconds. Find the distance covered by the particle in 1 seconds and also the acceleration at t = seconds. t : v : Evaluate 1 + x 1 use Simpson s rule? Give reasons. taking h = 0., using Trapezoidal rule. Can you 60. Evaluate to three decimals, dividing the range of x + x integration into 8 equal parts using Simpson s rule Evaluate sin x + cos x correct to two decimal places using 0 seven ordinates Find the value of log 3 from x 1 + x using Simpson s onethird rule with h = When a train is moving at 30 m/sec, steam is shut off and brakes are applied. The speed of the train per second after t seconds is given by Time (t) : Speed (v):

43 Using Simpson s rule, determine the distance moved by the train in 40 seconds. 64. Given e 0 = 1, e 1 =.7, e = 7.39, e 3 = 0.09, e 4 = 54.60, use 4 x Simpson s rule to evaluate e. Compare your result with 0 exact value. 65. A solid of revolution is formed by rotating about the x-axis, the area between x-axis, x = 0, x = 1 and the curve through the points (0, 1), (0.5, ), (0.5, ), (0.75, ) and (1, ). Find the volume of solid. 66. Evaluate 7 3 x log x taking 4 strips. 1 x 67. Evaluate e taking h = 0.05, using Trapezoidal rule Compute 1. 5 x Evaluate x by dividing the range into 8 equal parts. π 70. Evaluate π sin x e taking h =. 6 0 x 71. Evaluate e taking 6 intervals. 0 π 0 3 π 7. Calculate sin x taking h = From the following table, calculate y. 0 x : y :

44 1 74. Evaluate 0 by Weddle s rule, taking n = 6. 1 x 75. A curve passes through the points (1, 0.), (, 0.7), (3, 1), (4, 1.3), (5, 1.5), (6, 1.7), (7, 1.9), (8,.1), (9,.3). Find the volume of the solid generated by revolving the area between the curve, the x-axis and x = 1, x = 0 about the x-axis. 76. The velocity of a train which starts from rest is given by the following table, time being reckoned in minutes from the start and speed in miles per hour. Minutes : Miles/hr: Find the total distance covered in 0 minutes. 77. The velocity v of a particle moving in a straight line covers a distance x in time t. They are related as follows: x : v : Find the time taken to traverse the distance of 40 units. 78. Find the distance traveled by the train between am and 1.30 pm from the data given below: Time : 11.50am 1.00noon 1.10pm 1.0pm 1.30pm Speed in km/h : Evaluate 1.0 π 80. Evaluate 0 x taking h = sin θ dθ, using Simpson s rule taking six equal intervals. 44

45 81. Apply Simpson s rule to find the value of 1 + x 0 3 by dividing the range into 4 equal parts. 8. The speed of a train at various times are given by t (hour) : v (in km/h): Find the total distance covered. PART C 1. Find the polynomial equation y = f (x) passing through (5, 1335), (, 9), (0, 5), (-1, 33) and (-4, 145) using Newton s divided difference method.. Using Newton s divided difference formula, find the values of f (), f (8) and f (15) given the following table: x : f (x) : If y (0) = -18, y (1) = 0, y (3) = 0, y (5) = -48, y (6) = 0, y (9) = 13104, find y = f (x) using Newton s divided difference method. 4. Find the polynomial equation of degree four passing through the points (8, 1515), (7, 778), (5,138), (4,43) and (, 3) using Newton s divided difference method. 5. Find the pressure of steam at 14 o C using Newton s divided difference formula. Temp o C : Pressure kgf/cm : Obtain the value of log using divided difference formula, given log =.8156, log =.818, log =.8189, log =

46 7. Find y (x = ) using Newton s divided difference formula, from the table. x : y : From the data given below, find the value of x when y = 13.5, using Lagrange s formula. x : y : Find the age corresponding to the annuity value 13.6 using Lagrange s formula, given: Age (x) : Annuity value (y): Using Lagrange s formula find f (6) given, x : f (x) : The following table gives the values of the probability integral f (x) = x e x π 0 corresponding to certain values of x. For what value of x is this integral equal to 0.5? x : f (x) : Interpolate y at x = 5 using Lagrange s formula, given x : y : Find y (15) from the following table, using Lagrange s method. x :

47 y (x) : Use Lagrange s method to find x corresponding to y = 100, given x : y : Find y (1.50) using Lagrange s formula, given x : y : Evaluate x by using (a) Trapezoidal rule (b) Simpson s rule. 3 Verify your results by actual integration Evaluate the integral I = log x using Trapezoidal, Simpson s and Weddle s rules. 18. Evaluate I = x 6 e using (a) Trapezoidal rule (b) Simpson s rule (c) Weddle s rule. Also, check up by direct integration. π 19. By dividing the range into 10 equal parts, evaluate sin x by 0 Trapezoidal rule and Simpson s rule. Verify your answer with integration Evaluate 1 + x 0 by (a) Trapezoidal rule (b) Simpson s rule (c) Weddle s rule. Also check up the results by actual integration. 1. Compute the value of using Simpson s rule and Trapezoidal x 1 rule. Take h =

48 π. Calculate sin 0 x by dividing the interval into ten equal parts, using Trapezoidal rule and Simpson s rule. 3. Compute the value of (sin x log x + e x ) taking h = 0. and using Trapezoidal rule, Simpson s rule and Weddle s rule. Compare your result by integration. 4. The velocity v of a particle at distance S form a point on its path is given by the table below. S in metre : v m/sec : Estimate the time taken to travel 60 metres by using Simpson s one-third rule. Compare your answer with Simpson s 3/8 th rule and Weddle s rule. 5. Evaluate x 1 by taking n = 6 using (a) Trapezoidal rule (b) Simpson s one-third and three-eight rule (c) Weddle s rule Evaluate e 0 x by dividing the range into 4 equal parts using (a) Trapezoidal rule and (b) Simpson s one-third rule Evaluate e 1 x by taking h = 0.1 using Simpson s rule Calculate 0.5 e x 1 x by taking 5 ordinates by Simpson s rule. π 9. Evaluate 0 sin x x, dividing into six equal parts using Simpson s rule, Weddle s rule and Trapezoidal rule. 30. A river is 40 m wide. The depth d in metres at a distance x metres from one bank is given by the table below: x :

49 y : Find the cross-section of the river by Simpson s rule Evaluate 0 e x with 10 intervals by Trapezoidal and Simpson s methods. THINGS TO REMEMBER 49

50 _ UNIT 5 Solution of First Order Differential Equations of the form y = f PART A (x, y) 1. Write Euler s formula to solve a differential equation.. Write Taylor s series formula to solve a differential equation. 3. Write Modified Euler s formula to solve a differential equation. 4. Write Improved Euler s formula to solve a differential equation. 5. Write the iterative formula of nd order Runge-Kutta method. 6. Write the iterative formula of 3 rd order Runge-Kutta method. 7. Write the iterative formula of 4 th order Runge-Kutta method. PART B 50

51 8. Using Taylor s series method, find, correct to four decimal places, the value of y (0.1), given dy = x + y and y (0) = 1. dy 9. Using Taylor s series method, find y (0.1) given, = x + y, y (0) = Using Taylor s series method find y (0.1) given, y ' = x y 1, y (0) = Using Taylor s series method solve, y ' = x + y, y (1.8) = 0 for y 10 (). 1. Using Taylor s series method find y (0.1) given, y' = x y and y (0) = Using Taylor s series method find y (1.1) given, y' =x y and y (1) = Given y' = y and y (0) = 1, determine the values of y at x = 0.01 (0.01) 0.04 by Euler method. 15. Using Euler s method, solve numerically the equation, y ' = x + y, y (0) = 1, for x = 0.0 (0.) Solve numerically, y ' = y + e Improved Euler s method. x, y (0) = 0, for x = 0., 0.4 by 17. Compute y at x = 0.5 by Modified Euler method, given, y'= xy, y (0) = Given, y' = x y, y (0) = 1, find correct to four decimal places the value of y (0.1), by using Improved Euler method. 19. Using Modified Euler method, find y (0.), y (0.1) given, dy = x + y, y (0) = Use Euler s method to find y (0.4) given, y ' = xy, y (0) = 1. 51

52 y x 1. Use Improved Euler method to find y (0.1) given, y' =, y (0) y + x = 1.. Use Modified Euler method and obtain y (0.) given, dy = log( x + y), y (0) = 1, h = Find y (001) given, y ' = x + y, y (0) = 1, using Improved Euler method. 4. Find y (1.5) taking h = 0.5 given, y ' = y 1, y (0) = If y ' = 1+ y, y (0) = 1, h = 0.1, find y (0.4). 6. Find by Improved Euler to get y (0.), y (0.4) given, y (1) = 0.5. dy y x 3 = + x if 7. Find y (0.) by Improved Euler method, given y' = xy, y (0) = if h = Obtain the values of y at x = 0.1, 0. using RK method of second order for the differential equation y' = y, given y (0) = Obtain the values of y at x = 0.1, 0. using RK method of third order for the differential equation y' = y, given y (0) = 1. dy 30. Find y (0.) given, = xy, y (0) = 1 taking h = 0. by RK method of fourth order. 31. Find y (0.1), y (0.) given y' = x y, y (0) = 1 taking h = 0.1 using RK method of second order. 3. Find y (0.1), y (0.) given y' = x y, y (0) = 1 taking h = 0.1 using RK method of third order. PART C dy 33. Solve: = x + y, given y (1) = 0 and get y (1.1), y (1.) by Taylor series method. Compare your result with the explicit solution. 5